In the last post, I hope I explained how to find the Woods-Saxon energy levels for a given parameters. I just searched the best fit Woods-Saxon parameters to best fit the neutron single-particle energy for 209Pb. This is a double magic nucleus, and the there is no large fragmentation for the neutron excitation energy, thus, the outermost neutron in this nucleus can be considered as a good single-particle.

The best fit parameters are

The rms difference is just 78 keV!! For each level, the difference is not more than 50 keV!

So, how can we interpret this fitting result? It means, the energy levels can be well explained by WS mean field.

Recall that, the mean field actually included a lot things, it is the effective single particle potential that a nucleon is feeling. When we pushing the rms value to be minium. We are actually finding the best mean field. And the difference is due to the residual interaction.

But what contained in the mean field and residual interaction? The total Hamiltonian is

the mean field approach is to add an artificial potential

such that $latex H_{R} $ is minimum.

And by fitting energy levels using Woods-Saxon energy levels, we are basically doing the same thing! And the mean field is explicitly containing 2-body force, 3N-force and so on. So, can we say “because the Woods-Saxon can explain the energy level very well, the tensor force or other force is insignificant” ? the answer is no. Because the Woods-Saxon potential explicitly contains tensor force and other force.